The Role of Generalized Matrix Inverses in Markov Chains at HCMUS, Sep 12
 

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The Role of Generalized Matrix Inverses in Markov Chains

Jeffrey Hunter, Auckland University of Technology, Auckland, New Zealand. [Only registered and activated users can see links. Click Here To Register...]

14g Chiều Thứ Năm, 12/9, Phòng F207, 227 Nguyễn Văn Cừ, Q5.

Generalized matrix inverses play significant roles in solving for various key properties of finite, irreducible, Markov chains, in particular, the stationary distribution and the moments of the first passage time distributions. This arises from the observation that generalized matrices are used to solve systems of singular linear equations. In the context of Markov chains, we consider generalized matrix inverses of the singular Markovian kernel, I - P, where P is the transition matrix of the Markov chain.

We survey the application of generalized matrix inverses to such problems. We also establish that, under the aforementioned conditions, all generalized inverses of the Markovian kernel can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying Markov chain. Special sub-families include Meyer’s group inverse of I - P, Kemeny and Snell’s fundamental matrix of the Markov chain, and the Moore-Penrose g-inverse.